News (only records)
2014
March 19: A second CPAP-3 record with 10546 digits by David Broadhurst, PrimeForm, Primo.
February 21: New CPAP-3 record with 10546 digits by David Broadhurst, PrimeForm, Primo.
February 1: A third CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.
January 11: A second CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.

2013
December 17: New CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.
November 5: New CPAP-3 record with 10042 digits by Jens Kruse Andersen, Pierre Cami, Ken Davis, NewPGen, PrimeForm, Primo.
November 1: New CPAP-5 record with 1209 digits by David Broadhurst,
PrimeForm, Primo.
October 31: A third CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water,
PrimeForm, Primo.
October 27: A second CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water,
PrimeForm, Primo.
October 25: A second CPAP-4 record with 3021 digits by David Broadhurst, PrimeForm, Primo.
October 25: New CPAP-4 record with 3021 digits by David Broadhurst, PrimeForm, Primo.
October 23: A second CPAP-4 record with 3020 digits by David Broadhurst, PrimeForm, Primo.
October 21: New CPAP-4 record with 3020 digits by David Broadhurst, PrimeForm, Primo.
October 21: New CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water,
PrimeForm, Primo.
October 16: A second CPAP-4 record with 2575 digits by David Broadhurst, PrimeForm, Primo.
October 13: New CPAP-4 record with 2575 digits by David Broadhurst, PrimeForm, Primo.
October 12: A second CPAP-4 record with 2574 digits by David Broadhurst, PrimeForm, Primo.
October 10: New CPAP-4 record with 2574 digits by David Broadhurst, PrimeForm, Primo.

2012
December 10: New CPAP-4 record with 2148 digits by Jim Fougeron, Primo.

2008
December 8: A second CPAP-10 record with 93 digits by Manfred Toplic, CP10. 10 years after the first.

2007
December 26: New CPAP-7 record with 266 digits by Jens Kruse Andersen.
November 12: New CPAP-4 record with 1777 digits and record difference 2880 by Ken Davis, NewPGen, Primo.
October 1: CPAP-3 record for probable primes allowed with 10042 digits by Jens Kruse
Andersen, Ken Davis, NewPGen, PrimeForm.
January 7: New sections: The largest known CPAP-k for each
k and CPAP-2.

2006
No new records.

2005
March 19: New CPAP-9 record with 101 digits (same as before) by Hans Rosenthal & Jens Kruse
Andersen.
February 27: New CPAP-3 record with 7535 digits by David Broadhurst, François
Morain, FastECPP, PrimeForm.

2004
December 31: New CPAP-5 with record difference 2310 by Jim Fougeron, Primo.
December 3: New CPAP-4 with record difference 2310 by Jens Kruse Andersen,
PrimeForm, Primo.
December 1: New CPAP-4 with record difference 2004 by Jim Fougeron.
November 30: New record table with The largest known CPAP-k difference,
containing new records for CPAP-3, -5, -6 and -7 by Torbjörn Alm & Jens Kruse Andersen.
November 18: New CPAP-3 record with 7402 digits.
November 18: This news section is started (this page opened September 5 2003).

Introduction
A prime number is a natural number which only has the two divisors 1 and itself.
The first are 2, 3, 5, 7, 11.
An AP-k is any case of k primes in arithmetic progression, i.e. of the form p+d·n for some d
(the difference between the primes) and k consecutive values of n.
Example: 41 + 6n for n = 0, 1, 2, 3 gives the AP-4 41, 47, 53, 59.
See
Primes in Arithmetic Progression Records for the largest and smallest AP-k.

A CPAP-k is an AP-k where the k primes are consecutive, i.e. there are no other primes
between them. (CPAP can mean many other things outside
mathematics).
The AP-4 41, 47, 53, 59 is not a CPAP-4 because 43 is also prime. But 47, 53, 59 is a
CPAP-3. This page is only about CPAP-k.

A CPAP-k search often has two parts: Find an AP-k and then test whether the k primes are
consecutive. If the difference between the primes is small then it is sometimes possible
to make sure in advance that all intermediate numbers will be composite.

k# (called k primorial) is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7 = 210.
2# = 2, 3# = 6, 5# = 30, 7# = 210, 11# = 2310
The prime difference in an AP-k (and thus a CPAP-k) must be a multiple of k# to avoid factors
≤ k, assuming the primes in the AP-k are above k.
Avoiding intermediate primes in a CPAP-k becomes harder when the prime difference is big,
so many searches only try for difference k#.
A CPAP-6 has minimal difference 6# = 30 which is low in this context.
CPAP-7 to -10 all have minimal difference 10# = 7# = 210 which makes it harder.
However it is possible to make a guarantee against intermediate primes in a CPAP-7 larger than
around 190 digits.
x177 has been used for this.

A CPAP-11 would have minimal difference 11# = 2310. This seems extremely hard to find
and nobody has even tried as far as I know. With current methods it may take trillions of cpu
GHz years according to the people who found the first known CPAP-10.

It seems likely that there are infinitely many CPAP-k with prime difference c·k#, for all c
and k. You will be famous (among mathematicians anyway) by proving this, because the proof
would probably cover lots of other cases, e.g. the k-tuple conjecture. k=2 and c=1 gives the
twin prime conjecture, enough for fame.
Ben Green & Terence Tao presented a proof in 2004 that The
primes contain arbitrarily long arithmetic progressions, but their result is
not about consecutive primes.

Submissions
I would like to hear of all CPAP's which make one of the record tables. Please mail any
you find or know about. Say who should get credit and how the primes were proved.
The tables are not for numbers which are only prp's (probable primes). I have software to
prove prp's up to a few thousand digits. You can submit CPAP's consisting of prp's
but I want shared credit for performing proofs above 2000 digits.
If a CPAP was found with an expression involving a small
or big constant then please give the expression and constant, not just a decimal expansion of
the primes.

A link on the year of a record is to an announcement of that record.
A link on "Primo" was to Primo certificates of primality until 23
January 2009 where the website moved. Most of the certificates are currently
offline. They are available by email request.

Sexy primes are two primes separated by 6. This can be extended to sexy triplets and sexy
quadruplets, but not further due to divisibility by 5 - apart from the single
non-consecutive quintuplet (5, 11, 17, 23, 29). The definition does not require consecutive primes
but the records have it.

The minimal CPAP-k is currently only known for k<7.
After that the prime difference must be at least 210 and the minimal solution is probably
so large that an exhaustive search for it would be extremely hard. Heuristics (estimates
based on probability) indicate the
minimal CPAP-7 may have 22 or 23 digits. The smallest known is 32 digits:
19252884016114523644357039386451 + 210n, n=0..6

The need for at least 209 · 6 = 1254 composites in a CPAP-k with k>6
means it is much harder to find CPAP's with small primes than larger ones. The below table
shows the 3 smallest known CPAP-k when the minimal is unknown. There are only two
known CPAP-10.

Finding a CPAP-k with large difference is harder because more numbers
must be composite at the same time.
The table shows the single largest known CPAP-k difference (in bold) for each k. If more
than one CPAP is known with that difference then only the first found is shown.
Only the minimal difference k# = 210 has been found for CPAP-8, -9 and -10.

The following table shows current and old records, some from before the record category was
added here.
The first discovered CPAP-7, -8, -9 and -10 are all listed. I would guess the
first CPAP-7 was also the first CPAP-5 and -6 with difference 210, and that no larger
difference was known until the listed records.
Hans Rosenthal's
A probable CPAP-3 with d=17676 is not listed because no primality proofs were performed.

CPAP-2
CPAP-k usually assumes k > 2, but otherwise a CPAP-2 is by definition any pair of
consecutive primes. Other parts of this page do not include CPAP-2. There are
infinitely many primes and thus infinitely many CPAP-2.
Only proved primes are allowed on this page, so the top-10 CPAP-2 are the top-10 cases of
2 consecutive proved primes. In 2013 (and probably for decades before that) this is the
top-10 twin primes,
due to limitations in searched prime forms and known primality proving methods.
The largest is 3756801695685 · 2666669 ± 1 with 200700 digits, found in
2011 by Timothy D. Winslow, PrimeGrid,
TwinGen, LLR.
1 is not considered a prime, so the minimal CPAP-2 is 2, 3.
This is an exception to the rule that a CPAP-k difference must be a multiple of k#.
The exception is only possible because this CPAP-k starts with k. The only other
exception is the CPAP-3 with primes 3, 5, 7.
The largest known CPAP-2 difference is the largest known prime gap between 2 proved primes.
This is a gap of 1113106 between 18662-digit primes, found in 2013 by Pierre Cami,
Michiel Jansen, Jens K. Andersen, PrimeForm, Primo.

Credited programs
The primality proving program is only credited above 300 digits.CP09 was a program/project by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.
CP10 was by the same people as CP09.Primo (formerly Titanix) by Marcel Martin.VFYPR by Tony Forbes.PrimeForm by the OpenPFGW group with George Woltman.NewPGen by Paul Jobling.APSieve by Michael Bell and Jim Fougeron.Proth.exe by Yves Gallot.FastECPP by François Morain, Jens Franke, Thorsten Kleinjung and Tobias Wirth.TwinGen by David Underbakke.LLR by Jean Penné.APTreeSieve by Jens Kruse Andersen.